Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory.

Some of the alternative set theories are:

• the theory of semiset
  Semiset
  In set theory, a semiset is a proper class which is contained in a set.The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek . It is based on a modification of the von Neumann-Bernays-Gödel set theory; in standard NBG, the existence of semisets is...
  
  s
• the set theory New Foundations
  New Foundations
  In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
  
  
• Positive set theory
• Internal set theory
  Internal set theory
  Internal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be...
  
  

Specifically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka

Petr Vopenka

Petr Vopěnka is a Czech mathematician. In the early seventies, he established the Alternative Set Theory , which he subsequently developed in a series of articles and monographs...

 and his students. It builds on some ideas of the theory of semiset

Semiset

In set theory, a semiset is a proper class which is contained in a set.The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek . It is based on a modification of the von Neumann-Bernays-Gödel set theory; in standard NBG, the existence of semisets is...

s, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

 for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel

Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...

 (or ZF) set theory, in which the axiom of infinity

Axiom of infinity

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory...

 is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from Cantor

Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...

(ZF) finite sets and they are called infinite in AST.